Question: Samantha opened a savings account and deposited $\$8192$. The account earns $10\%$ in interest annually. She makes no further deposits and does not withdraw any money. In $t$ years, she has $\$25{,}710$ in this account. Write an equation in terms of $t$ that models the situation.
The strategy This problem involves an initial deposit of money increasing by $10\%$ every year. So, to find the amount of money in the account over time, we repeatedly multiply the original deposit, $\$8192$, by $1.10$. [Why?] Because of this, we know we can model the situation with an exponential expression of the form $a(1+r)^x$, where $a$ is $8192$ and $r$ is $0.10$. We now only need to find $x$, which represents the number of times the interest is calculated. Finding the exponent Since the interest is annual, this means it is calculated one time per year. Because $t$ represents the number of years in this case, our exponent is simply $t$. Writing an equation We can now replace $x$ in the original model with $t$. Therefore, the expression $8192\cdot(1.10)^t$ models the amount of money in the account after $t$ years. Since we know that in $t$ years, Samantha has $\$25{,}710$ in this account, we can set the above expression equal to $25{,}710$. $25{,}710= 8192\cdot(1.10)^{t}$ The answer An equation that models the situation is $8192\cdot (1.10)^t=25710$.